Necessary and Sufficient Consistency Conditions for Fanbeam Projections Along a Line

Clackdoyle, Rolf

doi:10.1109/tns.2013.2251901

Cited by 26 publications

(30 citation statements)

References 58 publications

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“…This initial condition is also related to the Helgason-Ludwig consistency condition, which is a global condition for the Radon transform [25, 42, 11]. One can see from (46) that the scaled 2D non-TOF Fourier data

\hat{p} (ω_{1}, ω_{2}, 0; u_{0}, 0) / \sqrt{1 + u_{0}^{2}}

is independent of u 0 when ω 1 = 0.…”

Section: Fourier-based Rebinning Equations Among Different Data Typesmentioning

confidence: 99%

Fourier rebinning and consistency equations for time-of-flight PET planograms

Defrise

Matej

et al. 2016

Inverse Problems

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Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

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\hat{p} (ω_{1}, ω_{2}, 0; u_{0}, 0) / \sqrt{1 + u_{0}^{2}}

is independent of u 0 when ω 1 = 0.…”

Section: Fourier-based Rebinning Equations Among Different Data Typesmentioning

confidence: 99%

Fourier rebinning and consistency equations for time-of-flight PET planograms

Defrise

Matej

et al. 2016

Inverse Problems

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“…(2) to recently published fanbeam DCCs for sources along a line. 5 Both formulas involve integration of an angular variable φ with weight tan n φ/cosφ, and both integrals simplify to polynomial expressions if the data are consistent with some density function. An explanation of the link between these formulas is given in the Sec.…”

Section: A Parallel Projectionsmentioning

confidence: 99%

“…5. Although the previous proof 5 can be applied to the DCCs given by Eq. (2), an independent demonstration is given below, that is more natural in the context of parallel beam backprojection.…”

Section: A Parallel Projectionsmentioning

confidence: 99%

“…For conventional fanbeam projections taken with a circular source trajectory, DCCs have been obtained by simply re-expressing the parallel H-L conditions using fanbeam variables. [21][22][23] More recently, fanbeam conditions for a linear source trajectory have been published, 5 as well as a description of the mathematical relationships between known fanbeam consistency conditions. 24 For most of the above-cited cases, 11,[16][17][18][19][21][22][23] the consistency conditions are expressed as mathematical equations to be satisfied by a complete set of projection measurements.…”

Section: Introductionmentioning

confidence: 99%

“…24 For most of the above-cited cases, 11,[16][17][18][19][21][22][23] the consistency conditions are expressed as mathematical equations to be satisfied by a complete set of projection measurements. In some cases, 5,16,17,20 the consistency equations can be applied to subsets of projections, provided the projections are not truncated. A well-known such example is the H-L condition of order zero, that any collection of ideal parallel projections will have the same projection sum, provided the projections are not truncated.…”

Section: Introductionmentioning

confidence: 99%

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Data consistency conditions for truncated fanbeam and parallel projections

Clackdoyle

Desbat

2015

Medical Physics

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Calibration‐free beam hardening reduction in x‐ray CBCT using the epipolar consistency condition and physical constraints

et al. 2019

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Background The beam hardening effect is a typical source of artifacts in x‐ray cone beam computed tomography (CBCT). It causes streaks in reconstructions and corrupted Hounsfield units toward the center of objects, widely known as cupping artifacts. Purpose We present a novel efficient projection data‐based method for reduction of beam‐hardening artifacts and incorporate physical constraints on the shape of the compensation functions. The method is calibration‐free and requires no additional knowledge of the scanning setup. Method The mathematical model of the beam hardening effect caused by a single material is analyzed. We show that the effect of beam hardening on the resulting functions on the line integral measurements are monotonous and concave functions of the ideal data. This holds irrespective of any limiting assumptions on the energy dependency of the material, the detector response or properties of the x‐ray source. A regression model for the beam hardening effect respecting these theoretical restrictions is proposed. Subsequently, we present an efficient method to estimate the parameters of this model directly in projection domain using an epipolar consistency condition. Computational efficiency is achieved by exploiting the linearity of an intermediate function in the formulation of our optimization problem. Results Our evaluation shows that the proposed physically constrained ECC2 algorithm is effective even in challenging measured data scenarios with additional sources of inconsistency. Conclusions The combination of mathematical consistency condition and a compensation model that is based on the properties of x‐ray physics enables us to improve image quality of measured data retrospectively and to decrease the need for calibration in a data‐driven manner.

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Necessary and Sufficient Consistency Conditions for Fanbeam Projections Along a Line

Cited by 26 publications

References 58 publications

Fourier rebinning and consistency equations for time-of-flight PET planograms

Fourier rebinning and consistency equations for time-of-flight PET planograms

Data consistency conditions for truncated fanbeam and parallel projections

Calibration‐free beam hardening reduction in x‐ray CBCT using the epipolar consistency condition and physical constraints

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